Regions of Stability for Limit Cycles of Piecewise Linear Systems Jorge M. Gonc ¸alves California Institute of Technology CDS, MC 107-81, Pasadena, CA 91125 jmg@cds.caltech.edu http://www.cds.caltech.edu/∼jmg/ Abstract— This paper starts by presenting local stability con- ditions for limit cycles of piecewise linear systems (PLS), based on analyzing the linear part of Poincar´ e maps. Local stability guarantees the existence of an asymptotically stable neighbor- hood around the limit cycle. However, tools to characterize such neighborhood do not exist. This work gives conditions in the form of LMIs that guarantee asymptotic stability of PLS in a reasonably large region around a limit cycle, based on recent results on impact maps and surface Lyapunov functions (SuLF). These are exemplified with a biological application: a 4 th –order neural oscillator, also used in many robotics applications like, for example, juggling and locomotion. I. INTRODUCTION Piecewise linear systems (PLS) are characterized by a finite number of linear dynamical models together with a set of rules for switching among these models. This cap- tures discontinuity actions in the dynamics from either the controller or system nonlinearities. Although widely used and intuitively simple, PLS are computationally hard and only recently there has been some interesting results. In the analysis of equilibrium points of PLS, [4], [7], [11] construct piecewise quadratic Lyapunov functions in the state space, and [2], alternatively, constructs Lyapunov functions on the switching surfaces (SuLF) associated with the system. Many PLS, however, have limit cycles. For a large class of relay feedback systems [1], one of the simplest PLS, there will be limit cycle oscillations. Such systems can be found in numerous applications like electromechanical sys- tems, simple models of dry friction, delta-sigma modulators, automatic tuning of PID regulators. Walking can also be seen as an oscillatory motion [12]. Walking robots are typically designed to walk at some predetermined velocity which is to be maintained even in the presence of external perturbations. In biology, oscillations appear in several applications like cell cycle [8], excitable cells like cardiac cells [8], circadian rhythms [3], and neural oscillators [9]. Neural oscillators are also used in many robotics applications like, for example, juggling [14] and bipedal locomotion control [5]. Here, the oscillator, composed of a PLS, is connected in feedback with the robot to induce oscillations. Local stability of limit cycles of PLS can easily be checked by linearizing the Poincar´ e map. However, global stability or even characterization of stability in regions around limit cycles cannot, in general, be checked or found. Simulations and experiments are typically the only way to have an idea about the stability of the system. [13] constructs local quadratic Lyapunov functions and then checks what regions of stability they guarantee. This check, however, is in general computationally hard and becomes impracticable for systems of order higher than 2. The results in [7] do not apply since Lyapunov functions cannot be constructed in the state space to prove stability of limit cycles. We then turn to construct Lyapunov functions on switching surfaces (SuLF). In the past, such construction seemed like an impossible task since impact maps, i.e., maps from one switch to the next, are, in general, nonlinear, multivalued, and not continuous. The work by [2], however, introduced a new and simple way to construct SuLF by simply solving a set of LMIs, and proving this way global stability of several classes of PLS. There are, however, many applications that do not require checking global stability, either because this is very expensive or maybe the system is not defined everywhere. Models of walking robots, for instance, are typically only defined in some region of the state space. This paper investigates the problem of constructing reasonably large regions of stability around a limit cycle. Trajectories starting in this region are guaranteed to converge asymptotically to the limit cycle. This is done in two steps. First, impact maps are found to be contractive in the large as possible set of switching times. Then, invariant ellipsoids on switching surfaces are characterized. The method is exemplified with a 4th–order neural oscillator [9]. The remainder of this paper is organized as follows. Section II describes the problem to be solved, and gives existence and local stability results for limit cycles of PLS. Section III presents the main result of this paper followed by an example in section IV. Proofs of all results can be found in section V. Finally, conclusions can be found in section VI. II. PROBLEM DESCRIPTION Piecewise linear systems (PLS) are characterized by a set of affine linear systems ˙ x = A i x + B i (1) where x ∈ IR n is the state, together with a switching rule i(x) ∈{1, ..., M} (2) that depends on present and possibly also on past values of x. By a solution of (1)-(2) we mean functions (x, i) satisfying (1)-(2), where i(t ) is piecewise constant. t is a switching time of a solution of (1)-(2) if i(t ) is discontinuous